The above narrative turns out to be the prototype for Fourier orthogonal series domains in
higher dimensions. In the past two decades, starting from the addition formula for classical orthogonal
polynomials on the unit ball [1]}, closed form formulas for reproducing kernels of orthogonal
polynomials have been discovered for several regular domains, including the unit ball, regular simplex,
cylinder, as well as the unit sphere with inner product defined by weighted integrals, which makes study
of the Fourier orthogonal series on these domain feasible; see, for example, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [1]}, [14]} and their references. For unbounded classical domains,
we refer to [15]} as well as to [16]}, [17]} for references on more recent works, which however
require techniques beyond our narrative.
